ORIGINAL_ARTICLEAutobiographical notesI was born in Zagreb (Croatia) on October 26, 1936. My parents were Regina (née Pavić) (April17, 1916, Zagreb–March 9, 1992, Zagreb) and Cvjetko Trinajstić (September 9, 1913, Volosko–October 29, 1998, Richmond, Australia).http://ijmc.kashanu.ac.ir/article_45087_dcb8952d03d7c0cf9d02f509573121d2.pdf2017-09-01T11:23:202018-02-22T11:23:2023125710.22052/ijmc.2017.64354.1248Chemical graph theorymathematical chemistryNanad TrinajsticN.Trinajstićtrina@irb.hrtrue1LEAD_AUTHORORIGINAL_ARTICLEGraphs with smallest forgotten indexThe forgotten topological index of a molecular graph $G$ is defined as $F(G)=\sum_{v\in V(G)}d^{3}(v)$, where $d(u)$ denotes the degree of vertex $u$ in $G$. The first through the sixth smallest forgotten indices among all trees, the first through the third smallest forgotten indices among all connected graph with cyclomatic number $\gamma=1,2$, the first through the fourth for $\gamma=3$, and the first and the second for $\gamma=4,5$ are determined. These results are compared with those obtained for the first Zagreb index.http://ijmc.kashanu.ac.ir/article_43258_0b9303cff3a4e0713b73cf33798147fb.pdf2017-09-01T11:23:202018-02-22T11:23:2025927310.22052/ijmc.2017.43258Forgotten topological indexUnicyclic graphsBicyclic graphsTricyclic graphsTetracyclic graphsPentacyclic graphsI.Gutmangutman@kg.ac.rstrue1University of Kragujevac, SerbiaUniversity of Kragujevac, SerbiaUniversity of Kragujevac, SerbiaAUTHORA.Ghalavandali797ghalavand@gmail.comtrue2University of KashanUniversity of KashanUniversity of KashanAUTHORT.Dehghan-Zadehta.dehghanzadeh@gmail.comtrue3University of KashanUniversity of KashanUniversity of KashanAUTHORA.Ashrafiijmc@kashanu.ac.irtrue4University of KashanUniversity of KashanUniversity of KashanLEAD_AUTHORORIGINAL_ARTICLEOn the first variable Zagreb index‎The first variable Zagreb index of graph $G$ is defined as‎ ‎\begin{eqnarray*}‎ ‎M_{1,\lambda}(G)=\sum_{v\in V(G)}d(v)^{2\lambda}‎, ‎\end{eqnarray*}‎ ‎where $\lambda$ is a real number and $d(v)$ is the degree of‎ ‎vertex $v$‎. ‎In this paper‎, ‎some upper and lower bounds for the distribution function and expected value of this index in random increasing trees (recursive trees‎, ‎plane-oriented recursive trees and binary increasing trees) are‎ ‎given‎.http://ijmc.kashanu.ac.ir/article_45113_2a098655d1e3c140f89c3276a97f9e78.pdf2017-09-01T11:23:202018-02-22T11:23:2027528310.22052/ijmc.2017.71544.1262First variable Zagreb index‎‎Random increasing‎ ‎trees‎‎Distribution function‎‎Expected valueK.Moradianrst.kazemi@gmail.comtrue1Department of Statistics, Islamic Azad UniversityDepartment of Statistics, Islamic Azad UniversityDepartment of Statistics, Islamic Azad UniversityAUTHORR.Kazemir.kazemi@sci.ikiu.ac.irtrue2Imam Khomeini international universityImam Khomeini international universityImam Khomeini international universityLEAD_AUTHORM.Behzadibehzadi.mh@gmail.comtrue3Department of Statistics, Islamic Azad UniversityDepartment of Statistics, Islamic Azad UniversityDepartment of Statistics, Islamic Azad UniversityAUTHORORIGINAL_ARTICLEComputing the additive degree-Kirchhoff index with the Laplacian matrixFor any simple connected undirected graph, it is well known that the Kirchhoff and multiplicative degree-Kirchhoff indices can be computed using the Laplacian matrix. We show that the same is true for the additive degree-Kirchhoff index and give a compact Matlab program that computes all three Kirchhoffian indices with the Laplacian matrix as the only input.http://ijmc.kashanu.ac.ir/article_48532_3d7984d6e1bc469b6c8dc075c7bf9610.pdf2017-09-01T11:23:202018-02-22T11:23:2028529010.22052/ijmc.2017.64656.1249Degree-Kirchhoff indexLaplacian matrixJ.Palaciosjpalacios@unm.edutrue1The University of New Mexico, Albuquerque, NM 87131, USAThe University of New Mexico, Albuquerque, NM 87131, USAThe University of New Mexico, Albuquerque, NM 87131, USALEAD_AUTHORORIGINAL_ARTICLEOn the spectra of reduced distance matrix of the generalized Bethe treesLet G be a simple connected graph and {v_1,v_2,..., v_k} be the set of pendent (vertices of degree one) vertices of G. The reduced distance matrix of G is a square matrix whose (i,j)-entry is the topological distance between v_i and v_j of G. In this paper, we compute the spectrum of the reduced distance matrix of the generalized Bethe trees.http://ijmc.kashanu.ac.ir/article_48533_93ac14d2ea7c08d919a6c30039ff1ded.pdf2017-09-01T11:23:202018-02-22T11:23:2029129810.22052/ijmc.2017.30051.1116Reduced distance matrixGeneralized Bethe TreespectrumA.Heydaria-heidari@iau-arak.ac.irtrue1Arak University of TechnologyArak University of TechnologyArak University of TechnologyLEAD_AUTHORORIGINAL_ARTICLEOn the second order first zagreb indexInspired by the chemical applications of higher-order connectivity index (or Randic index), we consider here the higher-order first Zagreb index of a molecular graph. In this paper, we study the linear regression analysis of the second order first Zagreb index with the entropy and acentric factor of an octane isomers. The linear model, based on the second order first Zagreb index, is better than models corresponding to the first Zagreb index and F-index. Further, we compute the second order first Zagreb index of line graphs of subdivision graphs of 2D-lattice, nanotube and nanotorus of TUC4C8[p; q], tadpole graphs, wheel graphs and ladder graphs.http://ijmc.kashanu.ac.ir/article_49784_8a12c9b7c5c6942e17923e06c1f223ee.pdf2017-09-01T11:23:202018-02-22T11:23:2029931110.22052/ijmc.2017.83138.1284Topological indexline graphsubdivision graphNanostructuretadpole graphBBasavanagoudb.basavanagoud@gmail.comtrue1KARNATAK UNIVERSITY DHARWADKARNATAK UNIVERSITY DHARWADKARNATAK UNIVERSITY DHARWADAUTHORS.Patilshreekantpatil949@gmail.comtrue2Karnatak UniversityKarnatak UniversityKarnatak UniversityAUTHORH. Y.Denghydeng@hunnu.edu.cntrue3LEAD_AUTHORORIGINAL_ARTICLEAnti-forcing number of some specific graphsLet $G=(V,E)$ be a simple connected graph. A perfect matching (or Kekul'e structure in chemical literature) of $G$ is a set of disjoint edges which covers all vertices of $G$. The anti-forcing number of $G$ is the smallest number of edges such that the remaining graph obtained by deleting these edges has a unique perfect matching and is denoted by $af(G)$. In this paper we consider some specific graphs that are of importance in chemistry and study their anti-forcing numbers.http://ijmc.kashanu.ac.ir/article_49785_63810586ce414d4cc8a97c6dbf37dcb3.pdf2017-09-01T11:23:202018-02-22T11:23:2031332510.22052/ijmc.2017.60978.1235Anti-forcing numberAnti-forcing setCorona productS.Alikhanialikhani@yazd.ac.irtrue1Yazd University, Yazd, IranYazd University, Yazd, IranYazd University, Yazd, IranLEAD_AUTHORN.Soltanineda_soltani@ymail.comtrue2Yazd UniversityYazd UniversityYazd UniversityAUTHORORIGINAL_ARTICLEOn the forgotten topological indexThe forgotten topological index is defined as sum of third power of degrees. In this paper, we compute some properties of forgotten index and then we determine it for some classes of product graphs.http://ijmc.kashanu.ac.ir/article_43481_12fee5261e77f9121b93d736c6102d79.pdf2017-09-01T11:23:202018-02-22T11:23:2032733810.22052/ijmc.2017.43481Zagreb indicesForgotten indexGraph productsA.Khaksarikhm.paper@gmail.comtrue1Department of Mathematics, Payame Noor University, Tehran, 19395 &ndash; 3697, I. R. IranDepartment of Mathematics, Payame Noor University, Tehran, 19395 &ndash; 3697, I. R. IranDepartment of Mathematics, Payame Noor University, Tehran, 19395 &ndash; 3697, I. R. IranAUTHORM.Ghorbanimghorbani@srttu.edutrue2Department of mathematics, Shahid Rajaee Teacher Training UniversityDepartment of mathematics, Shahid Rajaee Teacher Training UniversityDepartment of mathematics, Shahid Rajaee Teacher Training UniversityLEAD_AUTHOR